This paper supplements the book by \textit{E. M. Barth} and the author [From axiom to dialogue (1982; Zbl 0495.03001)] by considering the possibility that a dialogue may be non-cumulative, that is, that a particular sentence may be accepted or established at one stage of a dialogue, but not accepted (in dispute) at a later stage. The dialogue systems are shown to be equivalent to Fitch-style natural deduction systems for fragments of S4. As dialogue-logic, the paper is within the Lorenzen school, with emphasis on ''winning'' a dialogue and strict separation between the roles of opponent and proponent. The results do not seem applicable to the systems of other schools of dialogue logic, for example that of Hamblin [see \textit{J. Mackenzie}, ''No logic before friday'', Synthese 63, 329-341 (1985)].